Source code for sympy.functions.elementary.complexes

from __future__ import print_function, division

from sympy.core import S, Add, Mul, sympify, Symbol, Dummy
from sympy.core.exprtools import factor_terms
from sympy.core.function import (Function, Derivative, ArgumentIndexError,
    AppliedUndef)
from sympy.core.numbers import pi
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.core.expr import Expr
from sympy.core.relational import Eq
from sympy.core.logic import fuzzy_not
from sympy.functions.elementary.exponential import exp, exp_polar
from sympy.functions.elementary.trigonometric import atan2

###############################################################################
######################### REAL and IMAGINARY PARTS ############################
###############################################################################


[docs]class re(Function): """ Returns real part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function. Examples ======== >>> from sympy import re, im, I, E >>> from sympy.abc import x, y >>> re(2*E) 2*E >>> re(2*I + 17) 17 >>> re(2*I) 0 >>> re(im(x) + x*I + 2) 2 See Also ======== im """ is_real = True unbranched = True # implicitely works on the projection to C @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg is S.ComplexInfinity: return S.NaN elif arg.is_real: return arg elif arg.is_imaginary or (S.ImaginaryUnit*arg).is_real: return S.Zero elif arg.is_Matrix: return arg.as_real_imag()[0] elif arg.is_Function and isinstance(arg, conjugate): return re(arg.args[0]) else: included, reverted, excluded = [], [], [] args = Add.make_args(arg) for term in args: coeff = term.as_coefficient(S.ImaginaryUnit) if coeff is not None: if not coeff.is_real: reverted.append(coeff) elif not term.has(S.ImaginaryUnit) and term.is_real: excluded.append(term) else: # Try to do some advanced expansion. If # impossible, don't try to do re(arg) again # (because this is what we are trying to do now). real_imag = term.as_real_imag(ignore=arg) if real_imag: excluded.append(real_imag[0]) else: included.append(term) if len(args) != len(included): a, b, c = (Add(*xs) for xs in [included, reverted, excluded]) return cls(a) - im(b) + c
[docs] def as_real_imag(self, deep=True, **hints): """ Returns the real number with a zero imaginary part. """ return (self, S.Zero)
def _eval_derivative(self, x): if x.is_real or self.args[0].is_real: return re(Derivative(self.args[0], x, evaluate=True)) if x.is_imaginary or self.args[0].is_imaginary: return -S.ImaginaryUnit \ * im(Derivative(self.args[0], x, evaluate=True)) def _eval_rewrite_as_im(self, arg): return self.args[0] - S.ImaginaryUnit*im(self.args[0]) def _eval_is_algebraic(self): return self.args[0].is_algebraic def _sage_(self): import sage.all as sage return sage.real_part(self.args[0]._sage_())
[docs]class im(Function): """ Returns imaginary part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function. Examples ======== >>> from sympy import re, im, E, I >>> from sympy.abc import x, y >>> im(2*E) 0 >>> re(2*I + 17) 17 >>> im(x*I) re(x) >>> im(re(x) + y) im(y) See Also ======== re """ is_real = True unbranched = True # implicitely works on the projection to C @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg is S.ComplexInfinity: return S.NaN elif arg.is_real: return S.Zero elif arg.is_imaginary or (S.ImaginaryUnit*arg).is_real: return -S.ImaginaryUnit * arg elif arg.is_Matrix: return arg.as_real_imag()[1] elif arg.is_Function and isinstance(arg, conjugate): return -im(arg.args[0]) else: included, reverted, excluded = [], [], [] args = Add.make_args(arg) for term in args: coeff = term.as_coefficient(S.ImaginaryUnit) if coeff is not None: if not coeff.is_real: reverted.append(coeff) else: excluded.append(coeff) elif term.has(S.ImaginaryUnit) or not term.is_real: # Try to do some advanced expansion. If # impossible, don't try to do im(arg) again # (because this is what we are trying to do now). real_imag = term.as_real_imag(ignore=arg) if real_imag: excluded.append(real_imag[1]) else: included.append(term) if len(args) != len(included): a, b, c = (Add(*xs) for xs in [included, reverted, excluded]) return cls(a) + re(b) + c
[docs] def as_real_imag(self, deep=True, **hints): """ Return the imaginary part with a zero real part. Examples ======== >>> from sympy.functions import im >>> from sympy import I >>> im(2 + 3*I).as_real_imag() (3, 0) """ return (self, S.Zero)
def _eval_derivative(self, x): if x.is_real or self.args[0].is_real: return im(Derivative(self.args[0], x, evaluate=True)) if x.is_imaginary or self.args[0].is_imaginary: return -S.ImaginaryUnit \ * re(Derivative(self.args[0], x, evaluate=True)) def _sage_(self): import sage.all as sage return sage.imag_part(self.args[0]._sage_()) def _eval_rewrite_as_re(self, arg): return -S.ImaginaryUnit*(self.args[0] - re(self.args[0])) def _eval_is_algebraic(self): return self.args[0].is_algebraic
############################################################################### ############### SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ################ ###############################################################################
[docs]class sign(Function): """ Returns the complex sign of an expression: If the expresssion is real the sign will be: * 1 if expression is positive * 0 if expression is equal to zero * -1 if expression is negative If the expresssion is imaginary the sign will be: * I if im(expression) is positive * -I if im(expression) is negative Otherwise an unevaluated expression will be returned. When evaluated, the result (in general) will be ``cos(arg(expr)) + I*sin(arg(expr))``. Examples ======== >>> from sympy.functions import sign >>> from sympy.core.numbers import I >>> sign(-1) -1 >>> sign(0) 0 >>> sign(-3*I) -I >>> sign(1 + I) sign(1 + I) >>> _.evalf() 0.707106781186548 + 0.707106781186548*I See Also ======== Abs, conjugate """ is_finite = True is_complex = True def doit(self, **hints): if self.args[0].is_zero is False: return self.args[0] / Abs(self.args[0]) return self @classmethod def eval(cls, arg): # handle what we can if arg.is_Mul: c, args = arg.as_coeff_mul() unk = [] s = sign(c) for a in args: if a.is_negative: s = -s elif a.is_positive: pass else: ai = im(a) if a.is_imaginary and ai.is_comparable: # i.e. a = I*real s *= S.ImaginaryUnit if ai.is_negative: # can't use sign(ai) here since ai might not be # a Number s = -s else: unk.append(a) if c is S.One and len(unk) == len(args): return None return s * cls(arg._new_rawargs(*unk)) if arg is S.NaN: return S.NaN if arg.is_zero: # it may be an Expr that is zero return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if arg.is_Function: if isinstance(arg, sign): return arg if arg.is_imaginary: if arg.is_Pow and arg.exp is S.Half: # we catch this because non-trivial sqrt args are not expanded # e.g. sqrt(1-sqrt(2)) --x--> to I*sqrt(sqrt(2) - 1) return S.ImaginaryUnit arg2 = -S.ImaginaryUnit * arg if arg2.is_positive: return S.ImaginaryUnit if arg2.is_negative: return -S.ImaginaryUnit def _eval_Abs(self): if fuzzy_not(self.args[0].is_zero): return S.One def _eval_conjugate(self): return sign(conjugate(self.args[0])) def _eval_derivative(self, x): if self.args[0].is_real: from sympy.functions.special.delta_functions import DiracDelta return 2 * Derivative(self.args[0], x, evaluate=True) \ * DiracDelta(self.args[0]) elif self.args[0].is_imaginary: from sympy.functions.special.delta_functions import DiracDelta return 2 * Derivative(self.args[0], x, evaluate=True) \ * DiracDelta(-S.ImaginaryUnit * self.args[0]) def _eval_is_nonnegative(self): if self.args[0].is_nonnegative: return True def _eval_is_nonpositive(self): if self.args[0].is_nonpositive: return True def _eval_is_imaginary(self): return self.args[0].is_imaginary def _eval_is_integer(self): return self.args[0].is_real def _eval_is_zero(self): return self.args[0].is_zero def _eval_power(self, other): if ( fuzzy_not(self.args[0].is_zero) and other.is_integer and other.is_even ): return S.One def _sage_(self): import sage.all as sage return sage.sgn(self.args[0]._sage_()) def _eval_rewrite_as_Piecewise(self, arg): if arg.is_real: return Piecewise((1, arg > 0), (-1, arg < 0), (0, True)) def _eval_rewrite_as_Heaviside(self, arg): from sympy import Heaviside if arg.is_real: return Heaviside(arg)*2-1 def _eval_simplify(self, ratio, measure): return self.func(self.args[0].factor())
[docs]class Abs(Function): """ Return the absolute value of the argument. This is an extension of the built-in function abs() to accept symbolic values. If you pass a SymPy expression to the built-in abs(), it will pass it automatically to Abs(). Examples ======== >>> from sympy import Abs, Symbol, S >>> Abs(-1) 1 >>> x = Symbol('x', real=True) >>> Abs(-x) Abs(x) >>> Abs(x**2) x**2 >>> abs(-x) # The Python built-in Abs(x) Note that the Python built-in will return either an Expr or int depending on the argument:: >>> type(abs(-1)) <... 'int'> >>> type(abs(S.NegativeOne)) <class 'sympy.core.numbers.One'> Abs will always return a sympy object. See Also ======== sign, conjugate """ is_real = True is_negative = False unbranched = True
[docs] def fdiff(self, argindex=1): """ Get the first derivative of the argument to Abs(). Examples ======== >>> from sympy.abc import x >>> from sympy.functions import Abs >>> Abs(-x).fdiff() sign(x) """ if argindex == 1: return sign(self.args[0]) else: raise ArgumentIndexError(self, argindex)
@classmethod def eval(cls, arg): from sympy.simplify.simplify import signsimp from sympy.core.function import expand_mul if hasattr(arg, '_eval_Abs'): obj = arg._eval_Abs() if obj is not None: return obj if not isinstance(arg, Expr): raise TypeError("Bad argument type for Abs(): %s" % type(arg)) # handle what we can arg = signsimp(arg, evaluate=False) if arg.is_Mul: known = [] unk = [] for t in arg.args: tnew = cls(t) if isinstance(tnew, cls): unk.append(tnew.args[0]) else: known.append(tnew) known = Mul(*known) unk = cls(Mul(*unk), evaluate=False) if unk else S.One return known*unk if arg is S.NaN: return S.NaN if arg is S.ComplexInfinity: return S.Infinity if arg.is_Pow: base, exponent = arg.as_base_exp() if base.is_real: if exponent.is_integer: if exponent.is_even: return arg if base is S.NegativeOne: return S.One if isinstance(base, cls) and exponent is S.NegativeOne: return arg return Abs(base)**exponent if base.is_nonnegative: return base**re(exponent) if base.is_negative: return (-base)**re(exponent)*exp(-S.Pi*im(exponent)) return if isinstance(arg, exp): return exp(re(arg.args[0])) if isinstance(arg, AppliedUndef): return if arg.is_Add and arg.has(S.Infinity, S.NegativeInfinity): if any(a.is_infinite for a in arg.as_real_imag()): return S.Infinity if arg.is_zero: return S.Zero if arg.is_nonnegative: return arg if arg.is_nonpositive: return -arg if arg.is_imaginary: arg2 = -S.ImaginaryUnit * arg if arg2.is_nonnegative: return arg2 # reject result if all new conjugates are just wrappers around # an expression that was already in the arg conj = arg.conjugate() new_conj = conj.atoms(conjugate) - arg.atoms(conjugate) if new_conj and all(arg.has(i.args[0]) for i in new_conj): return if arg != conj and arg != -conj: ignore = arg.atoms(Abs) abs_free_arg = arg.xreplace({i: Dummy(real=True) for i in ignore}) unk = [a for a in abs_free_arg.free_symbols if a.is_real is None] if not unk or not all(conj.has(conjugate(u)) for u in unk): return sqrt(expand_mul(arg*conj)) def _eval_is_integer(self): if self.args[0].is_real: return self.args[0].is_integer def _eval_is_nonzero(self): return fuzzy_not(self._args[0].is_zero) def _eval_is_zero(self): return self._args[0].is_zero def _eval_is_positive(self): is_z = self.is_zero if is_z is not None: return not is_z def _eval_is_rational(self): if self.args[0].is_real: return self.args[0].is_rational def _eval_is_even(self): if self.args[0].is_real: return self.args[0].is_even def _eval_is_odd(self): if self.args[0].is_real: return self.args[0].is_odd def _eval_is_algebraic(self): return self.args[0].is_algebraic def _eval_power(self, exponent): if self.args[0].is_real and exponent.is_integer: if exponent.is_even: return self.args[0]**exponent elif exponent is not S.NegativeOne and exponent.is_Integer: return self.args[0]**(exponent - 1)*self return def _eval_nseries(self, x, n, logx): direction = self.args[0].leadterm(x)[0] s = self.args[0]._eval_nseries(x, n=n, logx=logx) when = Eq(direction, 0) return Piecewise( ((s.subs(direction, 0)), when), (sign(direction)*s, True), ) def _sage_(self): import sage.all as sage return sage.abs_symbolic(self.args[0]._sage_()) def _eval_derivative(self, x): if self.args[0].is_real or self.args[0].is_imaginary: return Derivative(self.args[0], x, evaluate=True) \ * sign(conjugate(self.args[0])) return (re(self.args[0]) * Derivative(re(self.args[0]), x, evaluate=True) + im(self.args[0]) * Derivative(im(self.args[0]), x, evaluate=True)) / Abs(self.args[0]) def _eval_rewrite_as_Heaviside(self, arg): # Note this only holds for real arg (since Heaviside is not defined # for complex arguments). from sympy import Heaviside if arg.is_real: return arg*(Heaviside(arg) - Heaviside(-arg)) def _eval_rewrite_as_Piecewise(self, arg): if arg.is_real: return Piecewise((arg, arg >= 0), (-arg, True)) def _eval_rewrite_as_sign(self, arg): from sympy import sign return arg/sign(arg)
[docs]class arg(Function): """ Returns the argument (in radians) of a complex number. For a positive number, the argument is always 0. Examples ======== >>> from sympy.functions import arg >>> from sympy import I, sqrt >>> arg(2.0) 0 >>> arg(I) pi/2 >>> arg(sqrt(2) + I*sqrt(2)) pi/4 """ is_real = True is_finite = True @classmethod def eval(cls, arg): if not arg.is_Atom: c, arg_ = factor_terms(arg).as_coeff_Mul() if arg_.is_Mul: arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else sign(a) for a in arg_.args]) arg_ = sign(c)*arg_ else: arg_ = arg if arg_.atoms(AppliedUndef): return x, y = re(arg_), im(arg_) rv = atan2(y, x) if rv.is_number: return rv if arg_ != arg: return cls(arg_, evaluate=False) def _eval_derivative(self, t): x, y = re(self.args[0]), im(self.args[0]) return (x * Derivative(y, t, evaluate=True) - y * Derivative(x, t, evaluate=True)) / (x**2 + y**2) def _eval_rewrite_as_atan2(self, arg): x, y = re(self.args[0]), im(self.args[0]) return atan2(y, x)
[docs]class conjugate(Function): """ Returns the `complex conjugate` Ref[1] of an argument. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number :math:`a + ib` (where a and b are real numbers) is :math:`a - ib` Examples ======== >>> from sympy import conjugate, I >>> conjugate(2) 2 >>> conjugate(I) -I See Also ======== sign, Abs References ========== .. [1] http://en.wikipedia.org/wiki/Complex_conjugation """ @classmethod def eval(cls, arg): obj = arg._eval_conjugate() if obj is not None: return obj def _eval_Abs(self): return Abs(self.args[0], evaluate=True) def _eval_adjoint(self): return transpose(self.args[0]) def _eval_conjugate(self): return self.args[0] def _eval_derivative(self, x): if x.is_real: return conjugate(Derivative(self.args[0], x, evaluate=True)) elif x.is_imaginary: return -conjugate(Derivative(self.args[0], x, evaluate=True)) def _eval_transpose(self): return adjoint(self.args[0]) def _eval_is_algebraic(self): return self.args[0].is_algebraic
class transpose(Function): """ Linear map transposition. """ @classmethod def eval(cls, arg): obj = arg._eval_transpose() if obj is not None: return obj def _eval_adjoint(self): return conjugate(self.args[0]) def _eval_conjugate(self): return adjoint(self.args[0]) def _eval_transpose(self): return self.args[0] class adjoint(Function): """ Conjugate transpose or Hermite conjugation. """ @classmethod def eval(cls, arg): obj = arg._eval_adjoint() if obj is not None: return obj obj = arg._eval_transpose() if obj is not None: return conjugate(obj) def _eval_adjoint(self): return self.args[0] def _eval_conjugate(self): return transpose(self.args[0]) def _eval_transpose(self): return conjugate(self.args[0]) def _latex(self, printer, exp=None, *args): arg = printer._print(self.args[0]) tex = r'%s^{\dagger}' % arg if exp: tex = r'\left(%s\right)^{%s}' % (tex, printer._print(exp)) return tex def _pretty(self, printer, *args): from sympy.printing.pretty.stringpict import prettyForm pform = printer._print(self.args[0], *args) if printer._use_unicode: pform = pform**prettyForm(u'\N{DAGGER}') else: pform = pform**prettyForm('+') return pform ############################################################################### ############### HANDLING OF POLAR NUMBERS ##################################### ###############################################################################
[docs]class polar_lift(Function): """ Lift argument to the Riemann surface of the logarithm, using the standard branch. >>> from sympy import Symbol, polar_lift, I >>> p = Symbol('p', polar=True) >>> x = Symbol('x') >>> polar_lift(4) 4*exp_polar(0) >>> polar_lift(-4) 4*exp_polar(I*pi) >>> polar_lift(-I) exp_polar(-I*pi/2) >>> polar_lift(I + 2) polar_lift(2 + I) >>> polar_lift(4*x) 4*polar_lift(x) >>> polar_lift(4*p) 4*p See Also ======== sympy.functions.elementary.exponential.exp_polar periodic_argument """ is_polar = True is_comparable = False # Cannot be evalf'd. @classmethod def eval(cls, arg): from sympy import exp_polar, pi, I, arg as argument if arg.is_number: ar = argument(arg) # In general we want to affirm that something is known, # e.g. `not ar.has(argument) and not ar.has(atan)` # but for now we will just be more restrictive and # see that it has evaluated to one of the known values. if ar in (0, pi/2, -pi/2, pi): return exp_polar(I*ar)*abs(arg) if arg.is_Mul: args = arg.args else: args = [arg] included = [] excluded = [] positive = [] for arg in args: if arg.is_polar: included += [arg] elif arg.is_positive: positive += [arg] else: excluded += [arg] if len(excluded) < len(args): if excluded: return Mul(*(included + positive))*polar_lift(Mul(*excluded)) elif included: return Mul(*(included + positive)) else: return Mul(*positive)*exp_polar(0) def _eval_evalf(self, prec): """ Careful! any evalf of polar numbers is flaky """ return self.args[0]._eval_evalf(prec) def _eval_Abs(self): return Abs(self.args[0], evaluate=True)
[docs]class periodic_argument(Function): """ Represent the argument on a quotient of the Riemann surface of the logarithm. That is, given a period P, always return a value in (-P/2, P/2], by using exp(P*I) == 1. >>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument >>> from sympy import I, pi >>> unbranched_argument(exp(5*I*pi)) pi >>> unbranched_argument(exp_polar(5*I*pi)) 5*pi >>> periodic_argument(exp_polar(5*I*pi), 2*pi) pi >>> periodic_argument(exp_polar(5*I*pi), 3*pi) -pi >>> periodic_argument(exp_polar(5*I*pi), pi) 0 See Also ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the Riemann surface of the logarithm principal_branch """ @classmethod def _getunbranched(cls, ar): from sympy import exp_polar, log, polar_lift if ar.is_Mul: args = ar.args else: args = [ar] unbranched = 0 for a in args: if not a.is_polar: unbranched += arg(a) elif isinstance(a, exp_polar): unbranched += a.exp.as_real_imag()[1] elif a.is_Pow: re, im = a.exp.as_real_imag() unbranched += re*unbranched_argument( a.base) + im*log(abs(a.base)) elif isinstance(a, polar_lift): unbranched += arg(a.args[0]) else: return None return unbranched @classmethod def eval(cls, ar, period): # Our strategy is to evaluate the argument on the Riemann surface of the # logarithm, and then reduce. # NOTE evidently this means it is a rather bad idea to use this with # period != 2*pi and non-polar numbers. from sympy import ceiling, oo, atan2, atan, polar_lift, pi, Mul if not period.is_positive: return None if period == oo and isinstance(ar, principal_branch): return periodic_argument(*ar.args) if isinstance(ar, polar_lift) and period >= 2*pi: return periodic_argument(ar.args[0], period) if ar.is_Mul: newargs = [x for x in ar.args if not x.is_positive] if len(newargs) != len(ar.args): return periodic_argument(Mul(*newargs), period) unbranched = cls._getunbranched(ar) if unbranched is None: return None if unbranched.has(periodic_argument, atan2, arg, atan): return None if period == oo: return unbranched if period != oo: n = ceiling(unbranched/period - S(1)/2)*period if not n.has(ceiling): return unbranched - n def _eval_evalf(self, prec): from sympy import ceiling, oo z, period = self.args if period == oo: unbranched = periodic_argument._getunbranched(z) if unbranched is None: return self return unbranched._eval_evalf(prec) ub = periodic_argument(z, oo)._eval_evalf(prec) return (ub - ceiling(ub/period - S(1)/2)*period)._eval_evalf(prec)
def unbranched_argument(arg): from sympy import oo return periodic_argument(arg, oo)
[docs]class principal_branch(Function): """ Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm. This is a function of two arguments. The first argument is a polar number `z`, and the second one a positive real number of infinity, `p`. The result is "z mod exp_polar(I*p)". >>> from sympy import exp_polar, principal_branch, oo, I, pi >>> from sympy.abc import z >>> principal_branch(z, oo) z >>> principal_branch(exp_polar(2*pi*I)*3, 2*pi) 3*exp_polar(0) >>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi) 3*principal_branch(z, 2*pi) See Also ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the Riemann surface of the logarithm periodic_argument """ is_polar = True is_comparable = False # cannot always be evalf'd @classmethod def eval(self, x, period): from sympy import oo, exp_polar, I, Mul, polar_lift, Symbol if isinstance(x, polar_lift): return principal_branch(x.args[0], period) if period == oo: return x ub = periodic_argument(x, oo) barg = periodic_argument(x, period) if ub != barg and not ub.has(periodic_argument) \ and not barg.has(periodic_argument): pl = polar_lift(x) def mr(expr): if not isinstance(expr, Symbol): return polar_lift(expr) return expr pl = pl.replace(polar_lift, mr) if not pl.has(polar_lift): res = exp_polar(I*(barg - ub))*pl if not res.is_polar and not res.has(exp_polar): res *= exp_polar(0) return res if not x.free_symbols: c, m = x, () else: c, m = x.as_coeff_mul(*x.free_symbols) others = [] for y in m: if y.is_positive: c *= y else: others += [y] m = tuple(others) arg = periodic_argument(c, period) if arg.has(periodic_argument): return None if arg.is_number and (unbranched_argument(c) != arg or (arg == 0 and m != () and c != 1)): if arg == 0: return abs(c)*principal_branch(Mul(*m), period) return principal_branch(exp_polar(I*arg)*Mul(*m), period)*abs(c) if arg.is_number and ((abs(arg) < period/2) == True or arg == period/2) \ and m == (): return exp_polar(arg*I)*abs(c) def _eval_evalf(self, prec): from sympy import exp, pi, I z, period = self.args p = periodic_argument(z, period)._eval_evalf(prec) if abs(p) > pi or p == -pi: return self # Cannot evalf for this argument. return (abs(z)*exp(I*p))._eval_evalf(prec)
def _polarify(eq, lift, pause=False): from sympy import Integral if eq.is_polar: return eq if eq.is_number and not pause: return polar_lift(eq) if isinstance(eq, Symbol) and not pause and lift: return polar_lift(eq) elif eq.is_Atom: return eq elif eq.is_Add: r = eq.func(*[_polarify(arg, lift, pause=True) for arg in eq.args]) if lift: return polar_lift(r) return r elif eq.is_Function: return eq.func(*[_polarify(arg, lift, pause=False) for arg in eq.args]) elif isinstance(eq, Integral): # Don't lift the integration variable func = _polarify(eq.function, lift, pause=pause) limits = [] for limit in eq.args[1:]: var = _polarify(limit[0], lift=False, pause=pause) rest = _polarify(limit[1:], lift=lift, pause=pause) limits.append((var,) + rest) return Integral(*((func,) + tuple(limits))) else: return eq.func(*[_polarify(arg, lift, pause=pause) if isinstance(arg, Expr) else arg for arg in eq.args]) def polarify(eq, subs=True, lift=False): """ Turn all numbers in eq into their polar equivalents (under the standard choice of argument). Note that no attempt is made to guess a formal convention of adding polar numbers, expressions like 1 + x will generally not be altered. Note also that this function does not promote exp(x) to exp_polar(x). If ``subs`` is True, all symbols which are not already polar will be substituted for polar dummies; in this case the function behaves much like posify. If ``lift`` is True, both addition statements and non-polar symbols are changed to their polar_lift()ed versions. Note that lift=True implies subs=False. >>> from sympy import polarify, sin, I >>> from sympy.abc import x, y >>> expr = (-x)**y >>> expr.expand() (-x)**y >>> polarify(expr) ((_x*exp_polar(I*pi))**_y, {_x: x, _y: y}) >>> polarify(expr)[0].expand() _x**_y*exp_polar(_y*I*pi) >>> polarify(x, lift=True) polar_lift(x) >>> polarify(x*(1+y), lift=True) polar_lift(x)*polar_lift(y + 1) Adds are treated carefully: >>> polarify(1 + sin((1 + I)*x)) (sin(_x*polar_lift(1 + I)) + 1, {_x: x}) """ if lift: subs = False eq = _polarify(sympify(eq), lift) if not subs: return eq reps = {s: Dummy(s.name, polar=True) for s in eq.free_symbols} eq = eq.subs(reps) return eq, {r: s for s, r in reps.items()} def _unpolarify(eq, exponents_only, pause=False): if isinstance(eq, bool) or eq.is_Atom: return eq if not pause: if isinstance(eq, exp_polar): return exp(_unpolarify(eq.exp, exponents_only)) if isinstance(eq, principal_branch) and eq.args[1] == 2*pi: return _unpolarify(eq.args[0], exponents_only) if ( eq.is_Add or eq.is_Mul or eq.is_Boolean or eq.is_Relational and ( eq.rel_op in ('==', '!=') and 0 in eq.args or eq.rel_op not in ('==', '!=')) ): return eq.func(*[_unpolarify(x, exponents_only) for x in eq.args]) if isinstance(eq, polar_lift): return _unpolarify(eq.args[0], exponents_only) if eq.is_Pow: expo = _unpolarify(eq.exp, exponents_only) base = _unpolarify(eq.base, exponents_only, not (expo.is_integer and not pause)) return base**expo if eq.is_Function and getattr(eq.func, 'unbranched', False): return eq.func(*[_unpolarify(x, exponents_only, exponents_only) for x in eq.args]) return eq.func(*[_unpolarify(x, exponents_only, True) for x in eq.args]) def unpolarify(eq, subs={}, exponents_only=False): """ If p denotes the projection from the Riemann surface of the logarithm to the complex line, return a simplified version eq' of `eq` such that p(eq') == p(eq). Also apply the substitution subs in the end. (This is a convenience, since ``unpolarify``, in a certain sense, undoes polarify.) >>> from sympy import unpolarify, polar_lift, sin, I >>> unpolarify(polar_lift(I + 2)) 2 + I >>> unpolarify(sin(polar_lift(I + 7))) sin(7 + I) """ if isinstance(eq, bool): return eq eq = sympify(eq) if subs != {}: return unpolarify(eq.subs(subs)) changed = True pause = False if exponents_only: pause = True while changed: changed = False res = _unpolarify(eq, exponents_only, pause) if res != eq: changed = True eq = res if isinstance(res, bool): return res # Finally, replacing Exp(0) by 1 is always correct. # So is polar_lift(0) -> 0. return res.subs({exp_polar(0): 1, polar_lift(0): 0}) # /cyclic/ from sympy.core import basic as _ _.abs_ = Abs del _